G89'2247 Lecture 11

1
G89.2247Lecture 11

• SEM analogue of General Linear Model
• Fitting structure of mean vector in SEM
• Numerical Example
• Growth models in SEM
• Willett and Sayer (1994) Examples

2
SEM as Analogue of General Linear Model

• Regression models can be used to estimate t tests
  and ANOVAs
• Groups are coded with dummy variables (0,1) or
  effect variables (-.5,.5)
• Regression parameters can be interpreted in terms
  of group means and differences between means
• Continuous covariates as well as interactions can
  be added to the model

3
Numerical Example of GLMDummy coded X
4
Taking Means into Account in SEM

• So far we have analyzed Variance Covariance
  Matrices, (First moments around the mean)
• Now we will analyze the general First moment
  matrix

5
New Information, New Parameters

• The general sums of squares matrix has p new
  pieces of information the variable means
• The new models will either
• Account for the means in a saturated model
• Or they will represent the means in a more
  parsimonious SEM model
• New ideas can be explored
• Harmony of covariance and means patterns
• Variance and covariance of contrasts

6
Path Representation of Means Models

• To fit the numerical example we need a constant
  term
• Y 32.48 12.86X e(Y)
• X .61 e(X)

e(Y)
e(X)
Y
X
1
7
Means in SEM Software

• In EQS the mean is the coefficient associated
  with a system variable called V999
• V999 represents the triangle
• In LISREL there are new Greek constant terms
• X tX LXx d
• Y tY LYh e
• h a Bh Gx z

8
Examples in Handout

• EQS Examples
• GLM version of t test
• GLM version of t test with covariate (ANCOVA)
• Covariate W is strongly related with group
  indicator X
• GLM version of t test with centered covariate
• Two group analysis with separate slopes

9
Means Models with Latent VariablesSaturated
Means Structure

• To date we have thought implicitly about latent
  variables as having mean zero. Let's be
  explicit. We have adjusted for manifest variable
  means.

10
Means Models with Latent VariablesInferred
Means Structure

• If the latent variable drives the means as well
  as the covariances, we get a different stronger
  model for the means of the manifest variables.

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Continuing with Examples in Handout

• EQS Examples
• Latent variable as covariate with mean zero
• Comparable to earlier example with centered
  covariate
• Latent variable as covariate with nonzero mean
• Comparable to earlier example with noncentered
  covariate

12
Latent Growth Models via SEM

• Suppose we had five repeated measures, spaced
  equally over time.
• An analysis of Y1, Y2, Y3, Y4, Y5 that uses only
  variance/covariances ignores trajectories.
• Willett and Sayer review SEM models that allow us
  to think about systematic linear growth.
• These models use mean structures.

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Example of Trajectories
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Latent Growth Models

• "Level 1" model Represents how Y changes over
  time points (Willett and Sayer notation)
• Yip p0p p1pti eip
• Suppose t1 0. Then p0p is the subject-specific
  intercept for the trajectory (the value of Y at
  time 1)
• The value p1p is the subject-specific slope of Y
  with a unit change of time.
• We will be able to study the covariation of the
  intercept and slopes in "Level 2" parts of the
  model
• Level 1 is between time, Level 2 is between person

15
Level 1 Models in SEM

• Diagram looks like confirmatory factor analysis,
  but the "loading" are fixed, not estimated.
• Within person processes are inferred from between
  person covariance patterns.

1
p0
p1
D2
D1
Y1
Y2
Y3
Y5
Y4
e1
e2
e3
e4
e5
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Level 2 and Level 1 Models
Group
1
p0
p1
D2
D1
Y1
Y2
Y3
Y5
Y4
e1
e2
e3
e4
e5
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Willett and Sayer Example

• 168 adolescents are measured at five points in
  time (ages 11, 12, 13, 14, 15)
• Outcome is Tolerance of Deviant Behavior
• Transformed with log function for analysis
• Questions
• Is there evidence that TDB is going up on
  average?
• Do youth vary in their slopes?
• Are there individual differences
• By gender?
• By early exposure to deviance?

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Three models

• Fitting random and average trajectories assuming
  that variances within person are stable (WS Model
  1)
• Fitting random and average trajectories assuming
  that variances within person are variable (WS
  Model 2)
• Fitting random and average trajectories and
  checking associations of slope and intercept with
  gender and exposure to deviance (WS Model 4)